Regarding the OP's request to know about dividing (and joining) trains there are lots of places where this happens on a daily basis (at least on weekdays - it's much rarer at weekends) in the south east. In fact there are a lot more, and it's a lot more complex, than I had realised:
Faversham (Dover and Ramsgate, Dover and Margate)
Ashford International (Ramsgate and Ramsgate, Ramsgate and Dover, Dover and Canterbury W)
Tunbridge Wells (Hastings and Hastings, Hastings and Ore)
Tonbridge (Ramsgate and Dover, Ramsgate and Ramsgate, Dover and Canterbury)
Redhill (Brighton and Tonbridge, Horsham / Tonbridge / Reigate, Three Bridges and Reigate, Portsmouth Harbour and Tonbridge, Southampton Central and Reigate, Chichester and Tonbridge, Horsham and Reigate, Tonbridge and Horsham, Tonbridge and Reigate)
Three Bridges (Brighton and Portsmouth Harbour)
Lewes (to Ore and Seaford)
Barnham (Southampton Central and Bognor Regis)
Haywards Heath (to Littlehampton and Hastings, Littlehampton and Ore, Littlehampton and Eastbourne, Worthing and Hastings, Worthing and Eastbourne, Hastings and Brighton, Eastbourne and Bognor Regis, Worthing and Ore)
Worthing (to Littlehampton and Portsmouth H)
Horsham (Portsmouth Harbour and Bognor Regis, Southampton Central and Bognor Regis, Portsmouth & Southsea and Bognor Regis)
I'm pretty sure I've missed something off in this area. I'm not including where some coaches are dropped off and terminated.
I'm pretty sure something used to divide at Paddock Wood a long time ago too.